1. Field of the Invention
The present invention relates to the field of bandpass filters, and more particularly, bandpass filters used in optical disk devices to capture a wobble signal in read head and write head circuits.
2. Description of the Related Art
Bandpass filters are often second-order or biquadratic. The bandwidth of such filters is directly the coefficient of the first-order term in the denominator polynomial of the second-order filter transfer function. Thus, linear adjustability of second-order filters is trivial and can be linearly controlled by the conductance of an element if an appropriate topology is chosen. Adjustment of bandwidth is desired in order to ensure that a particular signal--such as the wobble signal in an optical disk device--is accurately and fully passed by the filter for use by subsequent circuitry, while simultaneously increasing the filter's blocking of other unwanted signals (including noise, for example).
To accomplish linear adjustability in second-order filters, two conditions must be satisfied. First, the coefficient of the first order term in the denominator polynomial of the filter transfer function must be implemented by a conductance (not a resistance). This is necessary because the bandwidth is controlled by, and directly related to, conductance. Secondly, that conductance must be controlled by a voltage or current in a linear fashion.
MOSFETs are typically used in the positive feedback path of a second-order bandpass filter. The back-gate (or body) voltage of a MOSFET is used to control the conductance of the MOSFET. The prior art shows that conductance is linearly dependent on the controlling voltage as EQU G=(.mu..sub.p C.sub.ox /2)(W/L)(v.sub.G -V.sub.T)
In the prior art, the conductance versus body voltage, or back-gate voltage, relationship is not linear. Thus, the second condition for linear adjustability is not satisfied.
The prior art also does not teach high-order bandpass filters--e.g., bandpass filters, such as those comprising cascaded first-order or second-order filters or other arrangements that provide greater than second-order filter transfer functions--that are linearly adjustable. Even assuming that the bandwidth of a second-order filter is adjusted linearly, the overall bandwidth of a higher-order bandpass filter composed of two or more cascaded second-order filters would not be adjusted linearly. FIG. 1 shows two response curves 14 and 15 related to three prior art second-order filters cascaded together. The bottom set of curves 15 shows how the bandwidths of each individual second-order filter can be linearly adjusted. The top set of curves 14 shows the effect of adjusting the bandwidths of each cascaded second-order filter on the overall bandwidth of the high-order filter. Note that, in the top set of curves 14, overall bandwidth cannot be adjusted linearly by only adjusting the bandwidth of the individual second-order stages of the filter.
Moreover, the bandwidth of such a higher-order filter may not even be monotonous upon adjustment, that is, either always increasing or always decreasing but never both during adjustment in a particular direction. In other words, as the conductance of each second-order filter stage increased, the bandwidth might first go down but then up (or vice versa)--which is neither monotonous nor linear in behavior. FIG. 2 shows how the overall bandwidth of a high-order (i.e., sixth-order) filter is not linear in response to linear adjustment of its three cascaded second-order filters. In this Figure, the adjusted overall bandwidth of the high-order filter versus the bandwidth scaling factor (BWS), which is proportional to the bandwidths of individual second-order filters only, is shown by line 18.
Accordingly, prior art higher-order filters are deficient since, by varying linearly the bandwidths of each second-order bandpass filter block of a sixth-order bandpass filter, for example, overall bandwidth cannot be adjusted linearly.
Active Filter Models
Active filters, such as those usually using operational amplifiers (op amp), may generally be constructed by interconnecting commonly used elementary building blocks (also referred to herein as "models"). Examples of such known models include summers, integrators, and second-order biquads. These are well-known building blocks that are used in filter design.
In summers, various input signals are applied to the input of the summer. Generally, the output voltage is the negative sum of the input voltages when the input is fed into the negative input of the op amp. A summer has a resistance, but no capacitance, in the feedback path from the output of the op amp to the negative input of the op amp. Integrators are similar to summers, except that they have a capacitance in the feedback path. They may or may not contain a resistance in the feedback path. Their output voltage is the input voltage integrated over time. Biquads are usually formed using summers and/or integrators. The biquad is probably the most important basic building block used in the design of active filters. It typically realizes a transfer function such as: ##EQU1## where A.sub.1 is the bandwidth, A.sub.0 is the angular center frequency ((2.pi.f.sub.cntr).sup.2), s=j.omega. is the frequency parameter (where .omega.=2.pi.f is the radian frequency (rad/s)), and B=A.sub.1 multiplied by the gain at the center frequency is the gain of the filter.
A basic filter design approach is: (1) determine the transfer function of the desired filter, (2) obtain a signal flow graph (SFG) that realizes the transfer function of the desired filter, and then (3) obtain the corresponding filter circuit from that SFG. Consequently, the filter circuit will implement the target transfer function.
There may be many SFGs realizing the same transfer function. And, although they implement the same function, various circuits corresponding to different SFGs can behave differently. Differences that may exist include adjustability range, adjustability shape (i.e., the relationship of some parameters, such as center frequency or bandwidth or gain, to branch variables), sensitivity to variations in parameters and the like. This invention, in one part, involves the generation of signal flow graphs from which can be made linearly adjustable high order bandwidth filters. These signal flow graphs can be implemented in a wide variety of filter circuits.
It is also worthy to note that it is known that the transfer function of an SFG can be derived either by using Mason's gain formula or by solving a system of linear equations, each of which corresponds to sub-blocks of SFGs containing one internal node or output node.
Finally, in connection with generating SFGs in accordance with this aspect of the invention, it is noted that the linear equation represented by each sub-graph is unique. Accordingly, each exclusive sub-graph of an SFG yields an equation, and the entire SFG corresponds to a set of linear equations.
This filter design approach is described step by step, as follows, by way of an example. Assume the transfer function is, as described previously, expressed by the following: ##EQU2##
A signal flow graph that realizes this function is shown in FIG. 3, in which reference character 27 denotes the input to the filter, 34 denotes the output from the filter, reference characters 29-33 and 35 denote branches of the signal flow graph, and reference characters 28 and 36 denote internal voltage node variables. In a known manner, the SFG shown in FIG. 3 may essentially be formed by interconnecting two sub-graphs (i.e., where a branch otherwise present in a sub-graph is missing in FIG. 3, such a "missing branch" effectively corresponds to a branch with zero (0) valued branch transmittance). The SFG can be verified as having the desired transfer function by using Mason's gain formula.
The sub-graph in FIG. 4 is an example of a transfer function of a filter model comprising an integrator. The output depicted in this sub-graph corresponds to node 41 (N.sub.3), and corresponds to integration of the voltages applied to inputs 37 (N.sub.1) and 38 (N.sub.2). The specific manner in which the illustrated transfer function is implemented in a filter circuit is explained, below, in connection with the circuit shown in FIG. 5. In the SFG of FIG. 4, elements 39 (G.sub.2), and 40 (G.sub.3) represent conductances and loop 42 represents a feedback path from the output of op amp 41 to its input, which path contains a capacitor having a capacitance s and conductance G.sub.1.
Synthesis of a Circuit Implementing a SFG
The circuit shown in FIG. 5 (or sub-circuit, since it may be used to build the overall filter circuit) corresponds in performance to the sub-graph of FIG. 4. Inputs 43 (N.sub.1) and 44 (N.sub.2) in FIG. 5 correspond to inputs 37 (N.sub.1) and 38 (N.sub.2) in FIG. 4. Node 41 (N.sub.3) in FIG. 4 corresponds to op amp 48 in FIG. 5. Loop 42 in FIG. 4 corresponds to an RC circuit formed by a capacitor 49 and a resistor 50 in FIG. 5, which is connected between the output 51 of op amp 48 and the negative input 51A of the op amp 48. Conductance 39 (-G.sub.2) of FIG. 4 corresponds to resistor 45 in FIG. 5, and conductance 40 (G.sub.3) corresponds to an inverter 46 and a resistor 47 in FIG. 5. The voltage at output 51 in FIG. 5 corresponds to the integration of the voltages on inputs 43 and 44 of op amp 48 together with the voltage from RC circuit 49-50 going to node 51A.
Such sub-circuits can, in known fashion, be interconnected in the same way as the manner in which their corresponding sub-graphs are interconnected, thereby forming the entire SFG. In this manner, the filter circuit that corresponds to that overall SFG is constructed. It is also noteworthy that if, for a given SFG, the designer chooses a different type of sub-circuit (e.g., a Transconductance-C implementation, rather than a MOSFET-C implementation), a different filter circuit may be implemented. However, basic circuit properties, such as parameter-component value relationships (e.g., the linear relationship between the bandwidth and the MOSFET gate voltages, which is proportional to the conductance value of those MOSFETs), inherited from the topology (i.e., circuit interconnection form or architecture) remain the same for all of these circuits since they possess the same topology. In other words, they have been created from the same SFG.
Although the prior art has included these filter design techniques, they have not heretofore been used or combined in a manner that resulted in a higher-order bandpass filter characterized by linearly adjustable bandwidth. Accordingly, a need has existed relating to the design and implementation of such filters.